Moduli of curves and spin structures via algebraic geometry

Here we investigate some birational properties of two collections of moduli spaces, namely moduli spaces of (pointed) stable curves and of (pointed) spin curves. In particular, we focus on vanishings of Hodge numbers of type and on computations of Kodaira dimension. Our methods are purely algebro-geometric and rely on an induction argument on the number of marked points and the genus of the curves.